In injection molding, a very viscous molten polymer material, or resin, is forced to flow under high pressure into a mold cavity, essentially corresponding in shape to the desired part, until the resin completely fills the cavity. During the filling, the molten resin (or melt) begins to cool, eventually solidifying into the plastic part.
In addition to the portion of the mold cavity that reflects the shape of the desired part, the mold cavity may include runners (ducts through which molten resin flows from the nozzle of an injection molding press) ending in gates (determining injection locations and freezing sooner than interior resin in order to trap the resin inside the cavity). The term cavity is sometimes regarded as including runners and gates, but sometimes not. The remainder of this document applies equally well to either definition.
The physics of injection molding, especially in aspects related to viscous flow and cooling, is reasonably well known. Properties of molten resins needed for simulation can be either directly measured or extrapolated with a certain degree of accuracy.
Computer models for simulation of injection molding have been developed and implemented in sophisticated and expensive commercially available software. The intent of the software is to reduce trial and error in both part and mold design, as well as to aid in setting up parameters controlling the molding process. Model simulations can forecast the likelihood that, for example, the resin will not entirely fill the cavity before it freezes; regions of weakness (e.g, weld lines) may be present in the molded plastic part; air traps might occur causing unfilled bubbles in the molded part; or the molded part might have unacceptable sink marks, burns or other blemishes. After a computer model has been built for a cavity of a particular shape, it can be used many times to run simulations, possibly with different resins or process parameters. An unfavorable result from a simulation may lead to a redesign of the mold or of the part, or even to termination of the project. Failure can be avoided before a mold for the part is made. Several software products for simulation of injection molding are commonly used in industry, including products developed and marketed by Moldflow Corporation of Framingham, Mass.
Computer simulation of injection molding involves establishing a numerical flow model that expresses the balances of mass, forces and energy within the mold cavity. The equations representing these balances are called the governing equations of the flow. The equations can be solved to show how the pressure, velocity, and temperature variables will evolve over time within the various regions of the cavity. Predicting the time evolution of these flow variables throughout the spatial domain of the cavity is the essence of the simulation process.
The model is called a numerical model because the shape of the cavity and the flow variables at various times are represented by numbers within the computer, and also because various approximations may be made to the governing equations themselves and to their discrete analogs. The software code applies the discretized governing equations to the flow variables to advance them forward in simulated time, from the start of injection of the molten resin into the cavity and continuing usually until the resin completely freezes.
While the number of points within any given cavity is theoretically infinite, a numerical model only represents the flow with a finite number of flow variables assigned to a finite number of spatial locations. A numerically simulated flow is only an approximation to the behavior of a real resin injected into a real cavity. Many methods of reducing the theoretical governing equations (i.e., those that operate on continuous space and time variables) to discretized equations (representing reality with a finite number of spatial locations and distinct points in time) are known in the art. Underlying every numerical model is a discretization scheme, typically reflecting certain assumptions about the dependence of the flow variables (e.g., velocity, pressure, and temperature) upon spatial location and time. Most discretization schemes use discrete time steps. At each time step the balance equations are solved to update the flow variables in the region of the cavity currently filled with resin, and to determine the advance of the flow front.
Spatial discretization schemes in any flow model often fall into two broad categories. In finite difference modeling, a grid of points is imposed over the spatial domain being modeled. Flow variables are simulated at specific locations within each resulting gridcell. In finite element modeling, the spatial domain is subdivided into a mesh of small elements of certain predefined types typically having relatively simple shapes such as a tetrahedron or a prism. Mixed spatial discretization schemes are sometimes used; for example, finite elements might be used to represent some variables along an axis or in a plane, while levels or layers might be employed in the perpendicular direction, so that variables assigned to the layers are handled using finite difference methods.
Building the flow model may take considerable time, but even more time is often spent running simulations with the model. Generally, model accuracy increases when either the density of grid points or spatial elements is increased or the time step is shortened, but models having better resolution in space or time execute more slowly than their more coarsely discretized counterparts. (Refinement of the spatial and time discretization schemes in fluid flow are often coupled—refining the spatial grid may require a shorter time step.) Simulations that are slow to execute are costly, in terms of both consumption of computer resources and time elapsed on the wall clock while people are waiting to analyze the results. A choice of parameters determining space and time resolution of a discretization scheme necessarily involves tension between the need for greater accuracy and the need for faster execution.
One conceptually simple way to build a numerical flow model is to apply finite element methodology to a tetrahedral mesh filling the cavity. This approach may take advantage of existing algorithms for automated generation of tetrahedral meshes in three-dimensional (3D) domains with triangulated boundaries, as taught, for example, in U.S. Pat. No. 6,252,601. However, to capture strong variations of temperature, velocity and, especially, viscosity between the cavity walls, the mesh elements must be much smaller than the local part thickness. For this reason, 3D meshes suitable for simulation of injection molding produce very large flow models with hundreds of thousands, if not millions, of mesh elements and flow variables. The number of mesh elements can be especially large if numerous shallow features at the surface of the part, such as lettering or serration, have to be filled by the mesh. Solving very large models with computing resources typically available in the industry can be unacceptably slow and impractical. Thus, leaner (i.e., smaller and more efficient) flow models are often desired even though models based on 3D finite element methods are potentially more accurate.
Historically, simulation of injection molding was first developed for simple parts shaped as thin plates or shells with gradually varying thickness. The cavity might additionally include one ore more runners shaped usually as narrow round channels, either straight or curved. The flow of resin in two simple cavity geometries—thin more or less flat cavities and narrow round channels—lends itself to certain simplifications known as the Hele-Shaw (HS) approximations. Flow models based on these approximations do not resolve small details such as the fine structure of the flow front, but they can represent the overall picture of injection molding within such narrow channels and thin cavities reasonably well.
When HS models are used to simulate flow within thin and essentially flat cavities, simulation can be done considerably faster than with models based on exact governing equations. The efficiency of HS approximations in this geometry derives from the fact that due to high viscosity the flow is laminar and largely controlled by the proximate cavity walls, and the pressure variation in the direction normal (i.e., perpendicular) to the walls is much smaller than in the flow direction, which is parallel to the walls. The pressure field is therefore essentially two-dimensional, and can be approximated using 2D finite elements covering the midsurface of the thin cavity (i.e., an imagined surface equidistant from the opposite cavity walls). In a cylindrical channel, the other special geometry for which the HS model has been developed, the pressure can be approximated even more simply using 1D finite elements arranged along the channel axis. As a consequence, in both geometries, the discretization of the pressure field requires significantly fewer variables. This is particularly important because pressures at different spatial locations are tightly coupled together, as can be seen from the fact that placing an obstacle to flow in one location within the cavity nearly immediately affects the flow pattern everywhere. For this reason, the pressure field is usually determined at each time step by solving a large system of equations. The significant reduction in the number of pressure variables through the HS approximation greatly reduces simulation time.
As mentioned previously, in a thin quasi-flat cavity, a 2D mesh of finite elements with which pressure is approximated can be constructed covering the midsurface of the cavity. Each 2D mesh element can be extended by simply projecting it in the cross-midsurface direction to the bounding walls, and the resulting 3D element approximated by a prism. Because of the particular cavity geometry, prismatic elements are referred to as flat elements. Such a 3D perspective is needed because, unlike the pressure field, the temperature and velocity fields vary strongly between opposing cavity walls. Within flat elements, these fields can be described by their profiles Ti(z, t) and ui(z, t), where z is the coordinate in the direction perpendicular to the midsurface, t is time, and i is the index of an element.
In the even simpler case of a narrow round channel, pressure can be approximated by 1D mesh of elements constructed along the channel axis, and, similar to the flat case, each element can then be projected out perpendicularly to the channel axis. The resulting quasi-cylindrical element is approximated by a round cylinder. Within a cylindrical element, the temperature and velocity fields can be described by their radial profiles Ti(r, t) and ui(r, t).
The governing equations controlling pressure, temperature and velocity in a cavity, under the assumption that the cavity can be approximated using only flat and cylindrical elements, are discussed in detail in the book Flow Analysis of Injection Molds by Peter Kennedy (Hanser Gardner Publications, 1995). The book also discusses procedures for assembling numerical HS flow models in simple cavities.
Illustration of the Nature of HS Approximation
In general, the temperature at a particular point in the cavity is controlled by two physical processes, advection and heat conduction. Advection occurs when a fluid moves, carrying hot and cold spots along with the movement. Heat conduction is caused by the random motion of molecules internal to the fluid, and will occur whether or not the fluid is in motion. It is conduction that will gradually bring a hot molded plastic part to mold temperature. It should be noted that turbulent diffusion of heat, a third process important in some fluid flow contexts, is unimportant to injection molding because the melt is strongly viscous, resulting in laminar flow.
The nature of the simplified treatment of the flow in the two geometries for which HS equations have been derived in the prior art is illustrated here by considering the pure cooling stage, which starts after resin has filled the entire cavity and its motion has essentially stopped. At this point, the temperature of the fluid changes solely due to heat conduction.
In the case of a round runner, the HS equation describing heat conduction in a cylindrical element characterized by a radius R can be written as
                                                        ∂                              ∂                t                                      ⁢                          E              ⁡                              (                                  r                  ,                  t                                )                                              =                      -                          q              ⁡                              (                                  r                  ,                  t                                )                                                    ,                                  ⁢                  0          <          r          <          R                ,                            (        1        )            wherein E(r, t) is the thermal energy per unit length contained in the core region of radius r of the cylindrical element,E(r,t)=∫0rcT(r1,t)2πr1dr1,  (2)q(r, t), the heat flux per unit length carrying heat toward the wall through the surface of the core region, can be written according to Fourier's law as
                              q          ⁡                      (                          r              ,              t                        )                          =                              -            2                    ⁢          π          ⁢                                          ⁢          r          ⁢                                          ⁢          κ          ⁢                      ∂                          ∂              r                                ⁢                                    T              ⁡                              (                                  r                  ,                  t                                )                                      .                                              (        3        )            c and κ correspond, respectively, to volumetric heat capacity and conductivity. The HS approximation for heat conduction in the cylindrical element is based not only on the assumption that the temperature within the element depends on only one spatial coordinate, in this case on radius r, but also on the assumption that the energy exchange with neighboring elements due to heat conduction is negligibly small. Both assumptions are well-justified provided that the element belongs to a long and narrow channel, and is not located at one of its ends.
The equation for heat conduction in a flat element characterized by half-thickness R can be written in a form nearly identical to equations (1)-(3), but, in this geometry, r represents the distance from the midsurface. E(r, t) and q(r, t) should now be understood as the thermal energy and heat flux per unit area, respectively, of a core layer with half-thickness r. The only difference from the round channel case is in the coefficients, which, as should be expected, depend on the channel geometry. Namely, in equations (2) and (3) respectively, the coefficients 2πr1 and 2πr, which can be viewed as shape functions characterizing cylindrical geometry, in the case of a flat geometry are replaced by a constant.
The profiles of temperature and velocity can be discretized using their values in a stack of layers that are generally parallel to the walls. For the geometry approximated by flat and cylindrical elements, typical layers are shown schematically in FIG. 1b and FIG. 2b, respectively, which are described in the section entitled “Detailed Description of the Invention.” The number of layers is commonly selected to be between 6 and 15 depending on the compromise between accuracy and speed requirements. In a numerical HS flow model, the total number of temperature and velocity variables throughout the mold cavity is thus typically 12-30 times larger than the total number of pressure variables. However, the profiles of temperature and velocity can be computed locally, unlike the pressure distribution, which requires global solution due to the strong coupling previously mentioned. Even when advection, describing the energy transfer by resin flow, is taken into account, explicit methods, which are known to practitioners of numerical modeling to be computationally fast, can be used to solve for the temperature and velocity fields at each time step. This explains why a large number of temperature and velocity variables less adversely affects computational performance than a comparable number of pressure variables would.
Appendix A provides a simplified example of a numerical HS flow model for a thin cavity having a gradually varying thickness. Such a cavity is shown schematically in FIG. 2a and FIG. 2b. In summary of the preceding discussion and Appendix A, the construction of numerical HS flow models for flat and shell-like parts requires constructing a mesh on the midsurface of the cavity, and assigning appropriate thicknesses to the mesh elements. The models combine two-dimensional description of pressure with the set of 1D profiles of temperature and velocity within each element. Such surface with thickness models are often referred to as 2.5D models.
Application of the HS Approach to Other Geometries
The HS approach was also applied to building flow models for more complex parts such as ribbed parts as well as parts containing various shallow features. In ribbed parts, the midsurface branches at the intersections where ribs connect with each other or with the main shell. The classical HS approximations do not work near intersections, as the standard flat and cylindrical elements cannot approximate the local geometry. The same is true for edges and rough shallow features. Consequently, in such locations discretized HS type equations are specified on an ad hoc basis, most often using flat elements with artificially assigned effective thicknesses.
Early implementations of 2.5D modeling required labor-intensive involvement of highly skilled technicians in the semi-manual construction of branched 2D meshes and assigning effective geometric characteristics to elements as needed to create complete flow models. The scarcity of, and consequently high pay demanded by, such skilled technicians hindered application of 2.5D modeling to parts with more complex shapes, and increased the importance of developing an automated setup of 2.5D models to minimize the time and expense associated with injection molding simulations.
Attempts to generate 2.5D models by automating the construction and consequent meshing of midsurfaces had little or no success, partly because the necessary mathematical generalization of midsurface, the skeleton, is a complex geometric object for non-trivially shaped parts. However, an interesting roundabout solution, called DUALDOMAIN technology, was developed and implemented by Moldflow Corporation in some of its software products.
In particular, U.S. Pat. Nos. 6,096,088 and 6,704,693 describe methods of essentially 2.5D modeling that do not require construction of branched 2D midsurface meshes. Instead, pre-existing or purposely constructed 2D meshes on the cavity walls are employed. Flow is modeled on paired meshes on opposing cavity walls. Each flow variable at a point in the interior of the mold cavity is assigned, redundantly, to two model elements associated with matched mesh elements, one from each of two opposing walls near the point. DUALDOMAIN technology then attempts to eliminate this redundancy by ensuring that the matched elements are assigned the same thickness, and by forcing synchronization of flow variables in the matched elements during simulation. For parts with moderately complex shapes, DUALDOMAIN technology enables automated flow model generation and produces efficient and reasonably accurate flow models.
It appears, however, that more complex thin-walled parts, in particular parts with numerous shallow features, may still be problematic for DUALDOMAIN technology. For example, serration and other intricate features on a large area of the part surface may result in a massively incorrect element matching, as well as lead to erroneous thickness assignments. To avoid this, defeaturing of the part design is usually recommended. But defeaturing can be a labor-intensive operation on par with the semi-manual midsurface construction, especially if it has to be done after the part geometry is exported out of the CAD program in which it was designed. Also, handling intersections where two or more geometric features come together may still require assigning effective thicknesses to some model elements on a case by case basis.